Rational Smoothness and Kazhdan—Lusztig Theory

Abstract

In this chapter we describe the notion of rational smoothness à la Kazhdan—Lusztig (cf. [78]). Rational smoothness is intuitively an approximation to smoothness defined using cohomological criteria. This is a weaker notion than that of smoothness. In this chapter we gather the known characterizations of rational smoothness of Schubert varieties in terms of Kazhdan—Lusztig polynomials, Bruhat graphs, T-stable curves, Poincaré polynomials, and more. One can also obtain a lower bound for the dimension of the tangent space T(w, τ) from the combinatorial data in the Bruhat graph. Criteria for rational smoothness and smoothness are also given by Kumar, and for the classical groups criteria are also given in terms of pattern avoidance (see Chapters 7 and 8).